Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-x+4y &= 6 \\ 5x+5y &= 5\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-5$ and the bottom equation by $4$ $\begin{align*}5x-20y &= -30\\ 20x+20y &= 20\end{align*}$ Add the top and bottom equations. $25x = -10$ Divide both sides by $25$ and reduce as necessary. $x = -\dfrac{2}{5}$ Substitute $-\dfrac{2}{5}$ for $x$ in the top equation. $+ \dfrac{2}{5}+4y = 6$ $\dfrac{2}{5}+4y = 6$ $4y = \dfrac{28}{5}$ $y = \dfrac{7}{5}$ The solution is $\enspace x = -\dfrac{2}{5}, \enspace y = \dfrac{7}{5}$.